a mathematical structure that assigns data consistently to every open region of a topological space, satisfying two axioms: restriction and gluing
for a topological space $X$, a sheaf $\mathcal{F}$ assigns to each open set $U \subseteq X$ a set (or group, ring, module...) $\mathcal{F}(U)$ of sections, with restriction maps $\rho_{UV}: \mathcal{F}(U) \to \mathcal{F}(V)$ whenever $V \subseteq U$
the two axioms that distinguish a sheaf from a presheaf:
- locality — if two sections agree on every element of a cover, they are equal
- gluing — if sections on the pieces of a cover agree on all overlaps, they can be assembled into a unique global section
the sheaf condition is the formal statement that local consistency implies global coherence
on a knowledge graph, a sheaf assigns data to neighborhoods of particles — local semantic frames — such that wherever two neighborhoods overlap, their frames agree. the tri-kernel fixed point is a sheaf-theoretic object: the focus distribution is the unique global section consistent with every local diffusion, spring, and heat constraint simultaneously
knowledge topology acquires sheaf structure when every local assignment (what a neuron knows about its neighborhood) can be glued into a consistent global picture without contradiction — the definition of aligned collective focus
sheaf cohomology measures the obstruction to gluing: $H^1(\mathcal{F}) \neq 0$ means local sections cannot always be assembled globally. in a cybergraph, nonzero cohomology corresponds to topological inconsistencies in the knowledge structure — contradictions that no amount of additional linking within the current topology can resolve
a presheaf satisfies only the restriction maps, not the gluing axiom. every sheaf is a presheaf; not every presheaf is a sheaf
sheafification is the canonical procedure that forces any presheaf into the nearest sheaf — analogous to taking the completion of a metric space
in category theory, sheaves on a site (a category with a Grothendieck topology) are the objects of a topos — the categorical generalization of a topological space
see also: topology, knowledge topology, category theory, collective focus theorem, tri-kernel